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Emergent Symmetry in Brownian SYK Models and Charge Dependent Scrambling

In this work, we introduce a symmetry-based approach to study the scrambling and operator dynamics of Brownian SYK models at large finite $N$ and in the infinite $N$ limit. We compute the out-of-time-ordered correlator (OTOC) in the Majorana model without charge conservation and the complex model with charge conservation, and demonstrate that in both models taking the random average of the couplings gives rise to emergent symmetry structures. The random averaging exactly maps the operator dynamics of the Majorana model and the complex model to the imaginary time dynamics of an SU(2) spin and an SU(4) spin respectively, which become solvable in the large $N$ limit. Furthermore, the symmetry structure drastically reduces the size of the Hilbert space required to calculate the OTOC from exponential to linear in $N$, providing full access to the operator dynamics at all times for large finite $N$. In the case of the complex model with charge conservation, using this approach, we obtain the OTOC within each charge sector both numerically at finite $N$ and analytically in the large $N$ limit. We find that the time scale of the scrambling dynamics for all times and in each sector is characterized by the charge density. Furthermore, after proper rescaling, the OTOC corresponding to different finite charge densities collapses into a single curve at large finite $N$. In the large $N$ limit, the rescaled OTOCs at finite density are described by the same hydrodynamic equation as in the Majorana case.